Generation of Terahertz Radiation by Wave Mixing in Zigzag Carbon Nanotubes
ABSTRACT With the use of the semiclassical Boltzmann equation we have calculated a direct current (d.c) in undoped zigzag carbon nanotube (CN) by mixing two coherent electromagnetic waves with commensurate frequencies i.e and . This effect is attributed to the nonparabolicity of the electron energy band which is very strong in carbon nanotubes. We observed that the current is negative similar to that observed in superlattice. However if the phase shift lies between and there is an inversion and the current becomes positive. It is interesting to note that exhibit negative differential conductivity as expected for d.c through carbon nanotubes. This method can be used to generate terahertz radiation in carbon nanotubes. It can also be used in determining the relaxation time of electrons in carbon nanotubes Comment: 12pages,5figures
Generation of Terahertz Radiation by Wave Mixing in Zigzag Carbon Nanotubes
S.Y.Mensaha, S. S. Abukaria, N. G. Mensahb, K. A. Dompreha, A. Twuma and F. K. A.
aDepartment of Physics, Laser and Fibre Optics Centre, University of Cape Coast, Cape
bDepartment of Mathematics, University of Cape Coast, Cape Coast, Ghana
cInstitute of Mathematical Sciences, Accra, Ghana
*Corresponding author. aDepartment of Physics, Laser and Fibre Optics Centre, University of
Cape Coast, Cape Coast, Ghana
Tel.:+233 042 33837
E-mail address: firstname.lastname@example.org
(S. Y. Mensah)
With the use of the semiclassical Boltzmann equation we have calculated a direct current (d.c)
in undoped zigzag carbon nanotube (CN) by mixing two coherent electromagnetic waves with
commensurate frequencies i.e and
the electron energy band which is very strong in carbon nanotubes. We observed that the current is
negative similar to that observed in superlattice. However if the phase shift lies between and
. This effect is attributed to the nonparabolicity of
there is an inversion and the current becomes positive. It is interesting to note that exhibit negative
differential conductivity as expected for d.c through carbon nanotubes. This method can be used to
generate terahertz radiation in carbon nanotubes. It can also be used in determining the relaxation
time of electrons in carbon nanotubes.
PACS codes: 73.63.-b; 61.48.De
Keywords: Carbon Nanotubes; Harmonic Wave Mixing; Direct Current Generation; Terahertz
nonlinear medium can result in a product which has a zero frequency or static (d.c) electromagnetic
field. If such a nonlinear interference phenomenon happens in a semiconductor or semiconductor
device, then the static electric field may result into a d.c current or a dc voltage generation .
It is a known fact that coherent mixing of waves with commensurate frequencies in a
semiconductors [2-4]. Important among them is the heating mechanism where the nonlinearity is
related to the dependence of the relaxation constant on the electric field [4-7]. Goychuk and H nggi
 have also suggested another scheme of quantum rectification using wave mixing of an alternating
electric field and its second harmonic in a single miniband superlattice (SL). Their approach is based
on the theory of quantum ratchets and therefore the necessary conditions for the appearance of dc
include a dissipation (quantum noise) and an extended periodic system .
Infact, several mechanisms of nonlinearity could be responsible for the wave mixing in
Interesting to this paper is where the mechanism of nonlinearity is due to the nonparabolocity of
the electron energy spectrum. Notable among such materials are the superlattice (SL) and carbon
nanotubes (CNs). In superlattice the theory of wave mixing based on a solution of the Boltzmann
equation have been studied in [9-11]. In all these works, the situation where
were not studied directly. The first paper to study this situation
in SL can be found in . Recently this problem has been revisited in the following papers [1,13,
14] because of the interest it generates. We study this effect in zigzag carbon nanotubes.
This work will be organised as follows: section 1 deals with introduction; in section 2, we
establish the theory and solution of the problem; section 3, we discuss the results and draw
nanotubes (CNs) subjected to the electric mixing harmonic fields.
Following the approach of  we consider an undoped single-wall zigzag (n, 0) carbon
We further consider the semiclassical approximation in which the motion of -electrons are
considered as classical motion of free quasi-particles in the field of crystalline lattice with dispersion
law extracted from the quantum theory.
Considering the hexagonal crystalline structure of CNs and the tight binding approximation,
the dispersion relation is given as
for zigzag CNs 
transverse quasimomentum level spacing and is an integer. The expression for in Eq (2) is given
is the overlapping integral, is the axial component of quasimomentum, is
signs correspond to the valence and conduction bands, respectively. Due to the transverse
quantization of the quasi-momentum, its transverse component can take
is the C-C bond length and is Plank's constant divided by. The - and +
Unlike transverse quasimomentum
within the range
model is applicable to the case under consideration because we are restricted to temperatures and /or
voltages well above the level spacing , ie.
the temperature, is the charging energy. The energy level spacing
, the axial quasimomentum
, which corresponds to the model of infinitely long CN
is assumed to vary continuously
,where is Boltzmann constant, is
is given by
where is the Fermi speed and L is the carbon nanotube length .
Employing Boltzmann equation with a single relaxation time approximation.
Where e is the electron charge, is the equilibrium distribution function ,
is the distribution function, and is the relaxation time. The electric field
axis. The relaxation term of Eq (5) describes the electron-phonon scattering [18, 19] electron-
electron collisions, etc.
is applied along CNs
Expanding the distribution functions of interest in Fourier series as;
Where the coefficient,
is the factor by which the Fourier transform of the nonequilibrium distribution function differs
from its equilibrium distribution counterpart.
is the Dirac delta function,
is the coefficient of the Fourier series and
Substituting Eqs. (6) and (7) into Eq. (5) , and solving with Eq. (1) we obtain
where , , and is the Bessel function of the kth order.
Similarly, expanding in Fourier series with coefficients
and expressing the velocity as
We determine the surface current density as
and the integration is taken over the first Brillouin zone. Substituting Eqs. (7), (9) and (12) into (13)
and linearizing with respect to
and then averaging the result with respect to time we obtain the direct current subjected to
and as follows;
Subsequently will be represented by .
3. Results, Discussion and Conclusion
Using the solution of the Boltzmann equation with constant relaxation time τ, the exact expression for
current density in CNs subjected to an electric field with two frequencies
obtained after cumbersome analytical manipulation.
We noted that the current density is dependent on the electric field
, the frequency , the relaxation time and . To further understand how these parameters affect
and , the phase difference
, we sketched equation (14) using Matlab. Fig.1 represents the graph of on for
. We observed that the current decreases rapidly, reaches a minimum
, the current density rises monotonously while for
current rises and then oscillates. This indicates that at low frequency there is rectification while as at
high frequency some fluctuations occur. The rectification can be attributed to non ohmicity of the
and rises. For
carbon nanotube for the situation where it Bloch oscillates. The behaviour of the current is similar to
that observed in SL . See Fig. 2. In comparison with the result in  for
the ratio 33 which is quite substantial. We also observed a shift of the to the left with
increasing value of .
We sketched also the graph of against for . The graph also
displayed a negative differential conductivity. See Fig. 3. Interestingly like in SL as indicated in 
the current is always positive and has a maximum at the value
amplitude of the electric field
. It is worthwhile to note that
relaxation time of the electrons in the nanotube. e.g.
0.71 irrespective of the
can be used to determine the
so knowing you can determine . On
the other hand for typical value for of the frequency would be 1.2 THz.
Finally we sketched a 3 dimensional graph of the current against and See Fig .4. It is important
to note that when the phase shift lies between and there is an inversion. See Fig. 5.
In conclusion, we have studied the direct current generation due to the harmonic wave mixing in
zigzag carbon nanotubes and suggest the use of this approach in generation of THz radiation .The
experimental conditions for an observation of the dc current effect are practically identical to those
fulfilled in a recent experiment on the generation of harmonics of the THz radiation in a
semiconductor superlattice . This method can also be used to determine the relaxation time .
Fig. 1. is plotted against
Fig. 2. is plotted against
Zigzag Carbon nanotubes
Fig. 3. is plotted against
Zigzag Carbon nanotubes
Fig.4. is plotted against
for Zigzag Carbon nanotubes.
Fig. 5. is plotted against
. When the phase shift lies between and
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